论文标题
在dirichlet eta函数的非平凡零上
On the nontrivial zeros of the Dirichlet eta function
论文作者
论文摘要
We construct a two-parameter complex function $η_{κν}:\mathbb{C}\to \mathbb{C}$, $κ\in (0, \infty)$, $ν\in (0,\infty)$ that we call a holomorphic nonlinear embedding and that is given by a double series which is absolutely and uniformly convergent on compact sets in the entire复杂平面。功能$η_{κν} $收敛到dirichlet eta函数$η（s）$ as $κ\ to \ infty $。我们证明了至关重要的特性，对于足够大的$κ$，函数$η_{κν}（s）$可以表示为线性组合$η_{κν}（k）= \ sum_ {n = 0}^n = 0}^{\ infty} {\ infty} a_n（κ） $ a_ {n}（κ）\ in \ mathbb {r} $和$ a_ {0} = 1 $），实际上，我们拥有倒数公式$η（s）= \ sum_ {n = 0}^{\ infty}^{\ infty} b_n（κ）b_n（κ） $ b_ {n}（κ）\ in \ mathbb {r} $从$ a_ {n} $递归中获得）。通过使用这些结果以及ETA函数的功能关系，$η（s）=λ（s）η（1-s）$，我们绘制了Riemann假设的证明，在我们的环境中，该假设等同于以下事实。 $η（s^{*}）=η（1-s^{*}）= 0）$均位于关键行$σ^{*} = \ frac {1} {2} $上。
We construct a two-parameter complex function $η_{κν}:\mathbb{C}\to \mathbb{C}$, $κ\in (0, \infty)$, $ν\in (0,\infty)$ that we call a holomorphic nonlinear embedding and that is given by a double series which is absolutely and uniformly convergent on compact sets in the entire complex plane. The function $η_{κν}$ converges to the Dirichlet eta function $η(s)$ as $κ\to \infty$. We prove the crucial property that, for sufficiently large $κ$, the function $η_{κν}(s)$ can be expressed as a linear combination $η_{κν}(s)=\sum_{n=0}^{\infty}a_n(κ) η(s+2νn)$ of horizontal shifts of the eta function (where $a_{n}(κ) \in \mathbb{R}$ and $a_{0}=1$) and that, indeed, we have the inverse formula $η(s)=\sum_{n=0}^{\infty}b_n(κ) η_{κν}(s+2νn)$ as well (where the coefficients $b_{n}(κ) \in \mathbb{R}$ are obtained from the $a_{n}$'s recursively). By using these results and the functional relationship of the eta function, $η(s)=λ(s)η(1-s)$, we sketch a proof of the Riemann hypothesis which, in our setting, is equivalent to the fact that the nontrivial zeros $s^{*}=σ^{*}+it^{*}$ of $η(s)$ (i.e. those points for which $η(s^{*})=η(1-s^{*})=0)$ are all located on the critical line $σ^{*}=\frac{1}{2}$.
